Complex quantum states, more specifically a special kind of multipartite entangled quantum states referred to as cluster states, form the basis of a measurement-based model for quantum computation [1, 2] and for a related topological approach to quantum error correction [3]. These cluster states are composed of quantum bits, hereinafter referred to as qubits, where at least one qubits is entangled with more than one other qubits. The measurement-based quantum computation model implements algorithms using such cluster states, by means of single-qubit measurements. If the qubits are implemented using quantum optics, i.e. electromagnetic radiation or photons, they are referred to as optical cluster states.
Other multipartite quantum states such as the Greenberger-Horne-Zeilinger (GHZ) state [21] find applications in quantum metrology [22] and sensing, as the sensitivity of interferometers for example can be increased by a factor of the number of involved entangled photons. Entangled states can be assigned a so-called De Broglie wavelength [25], which can be much smaller than the one of a single particle having the same frequency, i.e. wavelength, and scales inversely with the amount of involved particles.
In a quantum optical approach, qubits, or other forms of quantum states, can be implemented as different polarization states, as different optical paths, as different time modes, or as different modes of a nonlinear resonator for example.
Multipartite entanglement makes cluster states different from multiple two-mode, or bipartite, entangled states generated for example by a single nonlinear optical interaction such as spontaneous parametric down-conversion (Sp-PDC) or spontaneous four-wave mixing (Sp-FWM).
Four-mode cluster states have been used to demonstrate quantum computation [1, 2, 4]. In order to increase the efficiency of these computations, both larger clusters as well as the simultaneous use of several clusters are required. This requires generating simultaneously multiplexed clusters with targeted size, which may be very challenging to achieve. With increasing complexity, an additional issue arises from the required scalability of the source with the final goal to achieve compact, low cost and stable devices, which can be multiplexed and packaged.
Until now, various different approaches to generate cluster states as well as other multipartite entangled states have been demonstrated, including coherent locking of multiple optical parametric oscillators, or superimposition of different nonlinear processes such as using pump directionality in a single nonlinear crystal [4], two different nonlinear crystals with a single excitation field [6], a single nonlinear crystal and multiple optical excitation fields [7], or synchronously pumping a single nonlinear crystal within a cavity [8].
Methods for generating multipartite entangled states so far rely on a second-order nonlinear interaction and, moreover, require very complex optical setups. Multiple quantum states have been generated for example by using an optical parametric oscillator below threshold with a wide nonlinear phase-matching bandwidth [6].
In addition to using quantum bits implemented using two-dimensional quantum systems such as photons' polarization [2, 4], quantum computation based on continuous variables such as amplitude and phase has also been implemented [5].
Systems for generating optical multipartite quantum states demonstrated so far rely on second-order nonlinear interactions and require large and complex optical setups. A significant limitation to second-order nonlinear optical crystals is that they require a non-centro symmetric crystal structure and are therefore not directly compatible for integration with silicon-based technology, i.e. are not CMOS-compatible [9]. This restriction drastically limits the possibility for compact integration and miniaturization of the corresponding devices. Another drawback is that these crystals are not directly compatible with fiber-based technology, meaning that all demonstrated cluster state sources are based on complex free-space optics, which may only be transferred out of the labs by overcoming immense difficulties.
As for third-order nonlinear systems, they suffer from a number of other drawbacks such as lower nonlinearity [10], and in cases when more than one excitation field is used, they are impaired by stimulated nonlinear processes, which are parasitic and dominant [10, 11] compared to the desired spontaneous processes needed for quantum state generation. In third-order nonlinear systems, only the generation of correlated and entangled photon pairs have been demonstrated so far, for example in optical fibers [12] as well as on chip in waveguides [13], photonic crystals [14], ring resonators [15] and in coupled resonator waveguides [16]. Recently, the generation of multiplexed correlated photon pairs distributed across a frequency comb has also been achieved on a chip [16]. However, up to now, third-order systems are limited to the generation of two-mode single correlated states.